Low rank canonical polyadic decomposition of tensors based on group sparsity

Abstract

Abstract-A new and robust method for low rank Canonical Polyadic (CP) decomposition of tensors is introduced in this paper. The proposed method imposes the Group Sparsity of the coefficients of each Loading (GSL) matrix under orthonormal subspace. By this way, the low rank CP decomposition problem is solved without any knowledge of the true rank and without using any nuclear norm regularization term, which generally leads to computationally prohibitive iterative optimization for largescale data. Our GSL-CP technique can be then implemented using only an upper bound of the rank. It is compared in terms of performance with classical methods, which require to know exactly the rank of the tensor. Numerical simulated experiments with noisy tensors and results on fluorescence data show the advantages of the proposed GSL-CP method in comparison with classical algorithms.

Dynamic brain effective connectivity analysis based on low-rank canonical polyadic decomposition: application to epilepsy

“…Therefore, the rank estimation of the tensor could be part of the overall optimization problem to fit the CPD model. Several techniques have been proposed to solve the rank estimation problem in the case of CPD such as the CORCONDIA [12], the minimum description length (MDL) [38], the Laplace Method [39], the cross-validation based method [13], the method for simultaneously estimating the rank and noise level [34], the quotient of differences in additional values [42] and the group-sparsity of the over-estimated loading matrices technique recently proposed in [28,55] which showed higher performance over the above mentioned techniques. Indeed, recent works [28,55] suggested to use a new group-sparsity of the over-estimated loading matrices of the considered tensor as a powerful way to optimally estimate the tensor rank.…”

“…Several techniques have been proposed to solve the rank estimation problem in the case of CPD such as the CORCONDIA [12], the minimum description length (MDL) [38], the Laplace Method [39], the cross-validation based method [13], the method for simultaneously estimating the rank and noise level [34], the quotient of differences in additional values [42] and the group-sparsity of the over-estimated loading matrices technique recently proposed in [28,55] which showed higher performance over the above mentioned techniques. Indeed, recent works [28,55] suggested to use a new group-sparsity of the over-estimated loading matrices of the considered tensor as a powerful way to optimally estimate the tensor rank. More precisely, authors in [28,55] showed that the mixed 2,1 -norm, as a mean to describe the group sparsity constraint, is a tighter complex envelop of the matrix rank function than the nuclear norm commonly used in this context.…”

“…Indeed, recent works [28,55] suggested to use a new group-sparsity of the over-estimated loading matrices of the considered tensor as a powerful way to optimally estimate the tensor rank. More precisely, authors in [28,55] showed that the mixed 2,1 -norm, as a mean to describe the group sparsity constraint, is a tighter complex envelop of the matrix rank function than the nuclear norm commonly used in this context. Furthermore, the 2,1 -norm defined in equation ( 1) is attractable from a numerical complexity point of view compared to the nuclear norm whose minimization relies essentially on an iterative computation of the singular value decomposition of the considered matrix.…”

“…In the present study, the PDC measure computed from iEEG signals is combined with a low rank tensor decomposition to provide an efficient way to track the evolution of brain connectivity along the epileptic seizure. More precisely, the PDC measure is combined with a CPD model but with a rank that, contrary to the strategy adopted in [45], is optimally computed as recently suggested in [28,55]. The PDC is computed on sliding time windows leading to an easy implementation and reasonable numerical complexity.…”

A Semi-Supervised Rank Tracking Algorithm For On-Line Unmixing Of Hyperspectral Images

“…TQÃ [14], whereQ is a diagonal matrix with the r th diagonal element defined by +δ . The parameter δ is a small value to avoid division by zero and .…”